非線性系統穩態反應積分觀察法之研究

A Study in Integration Method for Observing the Steady-State Responses of Non-Linear Systems

周建中(Chien-Chung Chou)

指導教授 ： 康淵 ； 張永鵬

中原大學/工學院/機械工程研究所/博士(2003年)

https://doi.org/10.6840/cycu200300668

摘要

在本研究中，提出了非線性系統穩態響應觀察法，被稱之為K積分法。此方法是以相平面上之相軌跡與原點之間的距離為積分函數，對時間積分。利用取值時間、積分區間的改變與K積分值的關係，可以對非線性系統多變的響應型態進行判別。
在研究中除了利用公式證明K積分之理論，為了闡述其使用之方法以及驗證其可用性，於文中以數值分析方法，分別探討了四種非線性方程式，包括Duffing方程式、van der Pol方程式及piecewisely linear方程式，以及一種2 DOF運動的磨蹭轉子系統。並且，將K積分法所得結果與傳統的穩態反應觀察法，包括頻譜分析、Poincaré截面法、關聯維數以及K2熵所得的結果進行比較，以探討各種方法於鑑別系統響應時之互補性與獨特性。
此外，針對Duffing方程式、van der Pol方程式以及piecewisely linear方程式受雙頻激振，當激振頻率比值為無理數時，以數值模擬得到系統之類週期響應，探討K積分法分析類週期響應的可行性。
本文也以實驗的方式，針對具有摩蹭元件以及油膜軸承支撐的轉子實驗模組進行量測，經由K積分法對這兩種非線性轉子的實際穩態反應訊號分析，與其他穩態反應觀察法比較，分析於真實機械系統響應識別上K積分法的優點與弱點。

關鍵字

類週期 ； 關聯維數 ； Poincaré截面法 ； K2熵 ； 積分觀察法

並列摘要

In this study, it proposes an observation method to identify the steady-state responses of non-linear system. The method is called K integration method, which integrating the distance of trajectories and origin in phase plane. The complex responses of non-linear system are identified by using the relation between the section times, integrated interval, and K integration value.
This study has proved that the theory of K integration method by formula. Several numerical examples, which include Duffing equation, van der Pol equation, piecewisely linear equation, and a kind of 2 DOF motion of rotor system with rubbing, are selected to illustrate the calculation and prove the usability. Furthermore, in the analysis, the K integration method is compared with spectrum analysis, Poincaré section method、correlation dimension and K2 entropy, which is based on analysis results to investigate the complementary and uniqueness in identifying the system responses.
In addition, using the numerical simulation to obtain the quasi-period responses of Duffing equation, van der Pol equation, and piecewisely linear equation subjected to double excitations which frequency ratio is irrational number. From the numerical results of system responses, which can investigate the feasibility in identifying the quasi-period responses of K integration method.
Also, this study measures the steady-state responses of two kinds rotor experimental models of a rotor kit, both set up are non-linear model with rubbing component or hydrodynamic bearing. Base on the signal analysis of steady-state responses for two real non-linear rotors by using K integration method. And the result by using other method for steady-state responses, which can be surveyed the advantage or disadvantage in the K integration method, which is used in real vibration response.

並列關鍵字

Poincaré section method ； K2 entropy ； quasi-period ； correlation dimension ； integration method

參考文獻

1.Ehrich, F. F., and O`Connor, J. J., “Stator Whirl with Rotors in Bearing Clearance,” Journal of Engineering for Power, Vol. 89(3), pp. 381-390, 1967.

2.Ehrich, F. F., “High Order Subharmonic Response of High Speed Rotors in Bearing Clearance,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 110(1), pp. 9-16, 1988.

3.Ehrich, F. F., “Some Observations of Chaotic Vibration Phenomena in High Speed Rotor Dynamics,” ASME Journal of Vibration and Acoustics, Vol. 113(1), pp. 50-57, 1991.

4.Ehrich, F. F., “Observations of Subcritical Superharmonic and Chaotic Response in Rotor Dynamics,” ASME Journal of Vibration and Acoustics, Vol. 114(1), pp. 93-100, 1992.

5.Ehrich, F. F., “Spontaneous Sidebanding in High Speed Rotor Dynamics,” ASME Journal of Vibration and Acoustics, Vol. 114(4), pp. 498-505, 1992.